Lesson 4 Water Wonderland Solidify Understanding

Learning Focus

Interpret the graphs and equations of functions.

Write equations for the graph of functions.

Combine two linear functions.

How does function notation connect to a graph?

What connections can be made with function notation and a story context?

How do I combine functions graphically and algebraically?

Open Up the Math: Launch, Explore, Discuss

Aly and Dayne work at a water park and are required to drain the water at the end of each month for the ride they supervise. Each uses a pump to remove the water from the small pool at the bottom of their ride. The graph given represents the amount of water in Aly’s pool, , and Dayne’s pool, , over time.

A red line labeled a(x) begins on the vertical axis about halfway up and descends to about the middle of the x-axis, where it ends. A blue line labeled d(x) begins slightly below the red line, descends to the x-axis, and ends slightly to the right of where the red line ends. Time (minutes)Amount of water (gallons)

1.

Make as many observations as possible with the information given in the graph.

Dayne figured out that the pump he uses drains water at a rate of and takes to drain.

2.

Write the equation to represent the draining of Dayne’s pool, . What does each part of the equation mean?

3.

Based on this new information, correctly label the graph shown.

4.

What values of make sense in this situation? Write the domain of the function that represents the amount of water in Dayne’s pool.

5.

What output values make sense in this situation? Write the range of the function that represents the amount of water in Dayne’s pool.

6.

Write the equation that represents the draining of Aly’s pool, . State the domain and range for the function, .

Equation:

Domain:

Range:

Based on the graph and corresponding equations for each pool, answer the following questions.

7.

When is ? What does this mean?

8.

Find . What does this mean?

9.

If , then . What does this mean?

10.

When is ? What does this mean?

This month, Aly and Dayne decided to work together by putting both pumps in the pool at the same time to drain their pools and created the equation:

When they wrote their equation, Aly and Dayne decided that they would only model the time when the pumps were in the pool so they didn’t have to worry about the short time that it took to move the pumps from one pool to the next.

11.

What does represent?

12.

Graph on the same axes as and .

A red line begins on the vertical axis about halfway up and descends to about the middle of the x-axis, where it ends. A blue line begins slightly below the red line, descends to the x-axis, and ends slightly to the right of where the red line ends. Time (minutes)222444666888101010121212141414161616181818202020222222242424262626282828303030323232Amount of water (gallons)100001000010000200002000020000300003000030000400004000040000500005000050000000

13.

Write the equation for the function using the graph you created. Compare this equation to the algebraic representation for finding the sum of the equations for and .

14.

Should the algebraic equation of be the same as the algebraic function created from the graph? Why or why not?

15.

Use both the graphical and the algebraic representation to describe key features of and explain what each feature means (each intercept, domain and range for this situation and for the equation, maxima and minima, whether or not is a function, etc.)

Domain:

What it means in this context:

Range:

What it means in this context:

Interval(s) of increase:

What it means in this context:

Interval(s) of decrease:

What it means in this context:

-intercept:

What it means in this context:

-intercept:

What it means in this context:

Rate of change:

What it means in this context:

Continuous? Why?

Function? Why?

16.

Explain why adding the two values of the -intercepts together in and can be used to find the -intercept in .

17.

Can a similar method be used to find the -intercept of ? Explain.

Ready for More?

Fletcher, another employee, decided to start draining the pools before Aly and Dayne showed up for work. When Aly and Dayne arrived, there were already fewer gallons of water to be drained. How does this change the equation for the amount of water Aly and Dayne must drain? How does this change the graph?

Takeaways

Function vocabulary and notation:

Lesson Summary

In this lesson we wrote the equation of two functions given graphically and with a story context. We connected the domain and range to the context and the graph. We deepened our understanding of function notation, learning to interpret the notation for graphs, tables, and equations. We learned that functions can be added together both graphically and algebraically.

Retrieval

Solve the equations for .

1.

2.

3.

List the key features of the function provided in the graph.

a piecewise function on a coordinate plane x–5–5–5555101010y–5–5–5000